ΔE 32.7, loose: different enough that you should pick on purpose. Gloomy Violet is a little darker, much more muted, cooler than Royal Purple.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 33). Gloomy Violet is a little darker, much more muted, cooler than Royal Purple.
The same difference applies the other way round: Royal Purple is a little lighter, much more saturated, warmer.
ΔE is the colour difference between our two swatch estimates: under about 6 is close, over about 15 is loose. Real paints also differ in finish, coverage and how they thin, so test before you commit.
See Royal Purple on the colour wheel See Gloomy Violet on the colour wheel
All matches: Royal Purple equivalents · Gloomy Violet equivalents